WMTL-代数中的蕴涵滤子及其应用

Implication Filters of WMTL-Algebras with Applications

MTL-代数是通过在剩余格中添加预线性公理得到的一类重要的基础逻辑代数,该文通过在剩余格中添加弱预线性公理建立了WMTL-代数,并对它的性质进行了细致讨论.首先,通过在剩余格中添加弱预线性公理的方法引入了WMTL-代数的概念,讨论了剩余格,WMTL-代数,MTL-代数的区别与联系;其次,在WMTL-代数中引入了蕴涵滤子的概念,并通过引入增强集给出了蕴涵滤子的等价刻画及蕴涵滤子的生成方法;第三,在WMTL-代数中引入了强同余关系的概念,给出了蕴涵滤子和强同余关系相互确定的方法:第四,证明了WMTL-代数的蕴涵滤子型商代数是WMTL-代数,蕴涵滤子型商代数是线性的当且仅当蕴涵滤子是素的;第五,在WMTL-代数L中证明了弱预线性公理的有限可积性质:(x→(x→y))~n∨(y→(y→x))~n=1(x,y∈L,n∈N_+);最后,证明了WMTL-代数中不含一个特定元素(不属于给定蕴涵滤子)的素蕴涵滤子的存在性,并证明了满足条件[1)={1}的WMTL-代数可以嵌入到其上所有素蕴涵滤子型商代数的乘积代数之中.

MTL-algebra,which is a kind of important basic logic algebra proposed by Esteva and G del through adding axiom of pre-linearity:(x→y)∨(y→x)=1(x,y∈L)to a residuated lattice L,is generalized to WMTL-algebra in the present paper,and the properties of WMTL-algebras are investigated in depth.Firstly,the concept of WMTL-algebra is proposed by adding the axiom of weak pre-linearity:(x→(x→y))∨(y→(y→x))=1(x,y∈L)to a residuated lattice L,the relationships among residuated lattice,WMTL-algebra,MTL-algebra are given,and differences among them are showed by constructing corresponding examples;Secondly,the concepts of implication filter,weak implication filter,MP filter,and*filter are proposed in a WMTL-algebra,the relationships among them are discussed.It is proved that in a WMTL algebra,a weak implication filter is equivalent to an MP filter,and that an MP filter containing strengthening set D={x→x 2|x∈L}of the WMTL algebra is equivalent to an implication filter,and a method of generating implication filter from a subset of a WMTL algebra is given with the help of strengthening set;Thirdly,the concept of strengthened congruence relation is put forward in a WMTL-algebra L,which is a usual congruence relation on the WMTL-algebra L such that for x∈L,x→x 2 is related to 1,it is proved that in a WMTL algebra,there is an one to one correspondence between implication filters and strengthened congruence relations,the methods of the mutual conversion between implication filters and strengthened congruence relations are given;Fourthly,the concept of prime implication filter is introduced to WMTL algebras,it is proved that the quotient algebra of a WMTL-algebra determined by the strengthened congruence relation induced by a implication filter is also a weak MTL-algebra,and that the quotient algebra determined by the strengthened congruence relation is linear if and only if the inducing implication filter is prime;Fifthly,using the mathematical induction,the property of finite product of weak pre-linearity axiom in WMTL-algebras is proved which says that in a WMTL-algebra L holds the fundamental equation:(x→(x→y))n∨(y→(y→x))n=1(x,y∈L,n∈N+);Finally,by utilizing the property of finite product of weak pre-linearity axiom in WMTL-algebras,it is proved that in a WMTL-algebra,for a given implication filter,there is a corresponding prime implication filter which contains the given implication filter and which does not contain a given point(not belonging to the given implication filter),then the conditional embedding theorem of a WMTL algebra is proved,which says that a WMTL-algebra satisfying the condition[1)={1}can be embedded in the product algebra of a family of linear quotient algebras of the WMTL-algebra determined by the strengthened congruence relations induced by corresponding prime implication filters.